My favourite is Oda's Strong Factorization Conjecture:

Can a proper, birational map between smooth toric varieties be factored as a composition of a sequence of smooth toric blow-ups followed by a sequence smooth toric blow-downs?

Note that if you are allowed to intermingle the blow-ups and blow-downs (the weak version) it has been proved. In fact, it was proved for general varieties in characteristic 0 *using* the toric case:

Torification and Factorization of Birational Maps. Abramovich, Karu, Matsuki, Wlodarczyk.

A conjectural algorithm for computing toric strong factorizations can be found in the following arXiv article:

On Oda's Strong Factorization Conjecture. Da Silva, Karu.

ConjectureEvery ample divisor on asmoothtoric variety is very ample and induces a projectively normal embedding. Is that right? $\endgroup$`question'' rather than`

conjecture'' as it doesn't seem like all experts believe it.) $\endgroup$2more comments